I imagine it's like Adolf Hitler. No one minded being named Grima or Wormtounge until after Grima Wormtounge.

Wormtongue was his nickname, presumably given after he was hired as councilor.

"I thought 'wormtongue' meant he would... well, it doesn't matter now."

I think I remember reading that it was supposed to be descended from wyrmtongue as in dragon talkers and grima's ancestors were revered dragon speakers, which is why he has such a prominent role in the king's chamber.